In  we prove that the group Gn of automorphisms of Fn which take every element of a fixed basis to a conjugate of itself has a regular language of normal forms. (In another guise, Gn appears as the group of motions of n circles in the 3-space and is of some importance in modern physics [BL], [K].) This is a part of an ongoing research motivated by the recent results showing that Aut(Fn) is not (bi)automatic (Bridson-Vogtmann [BV]), while its important subgroups, like the braid groups and the mapping class groups of surfaces are automatic (Thurston [E+], Mosher [M]). It is still to be seen if the methods employed in  can be extended to provide an answer to the questions of whether Gn is automatic, and whether Aut(Fn) has a regular language of normal forms.
In , we prove that the cohomology ring of the above group Gn has a simple presentation, as conjectured by Brownstein and Lee in [BL]: the generators and relators and . The proof uses the description of abelian subgroups of Gn given in , the construction of a constructible complex on which Gn acts with abelian cell stabilizers, and the spectral sequence associated with the action. The proof also gives a description of a free basis of every group Hq(Gn); it is in a bijective correspondence with the set of rooted forests on n vertices which have n-q components. A classical enumeration formula of Cayley implies then that the Poincaré polynomial of Gn is (1+nt)n-1.